On Vesentini’s Stone type theorem for one-parameter groups of non-linear automorphisms in Hilbert space

L.L. STACHÓ

Abstract.

AMS 2000 classification: 47D03, 32H15, 46G20.

1. Introduction, main results

Notations

E Banach space with predual E_{∗},

K bounded open convex subset in E with closure \overline{K},

𝔐 := \{matrices \left[{A b \atop c_{∗} d}\right] where A ∈ L(E), b ∈ E, c_{∗} ∈ E_{∗}, d ∈ ℂ\}, for A ∈ 𝔐,

Ψ_{A}(x) := (c_{∗}x + d)^{−1}(Ax + b) with dom(Ψ_{A}) = {x ∈ E : c_{∗}x + d ≠ 0}

Lemma 2.1. Assume K is a compact topological space and let [f_{t} : t ∈ ℝ_{+}] be a oneparameter semigroup of continuous maps K → K admitting fixed points such that all the functions t ↦ f_{t}(x) are continuous. Then also ⋂_{t∈ℝ_{+}} Fix(f_{t}) ≠ ∅.

Proof. Consider any parameter t > 0 and a point x ∈ Fix(f_{t}). For n = 1, 2, … recursively we have f_{nt}(x) = f_{t}(f_{(n−1)t}(x)) = x. Thus Fix(f_{nt}) ⊃ Fix(f_{t}) ≠ ∅ (t ∈ ℝ_{+}, n = 1, 2, …). From the continuity of the maps f_{t} it follows that Fix(f_{1/n!}) (n = 1, 2, …) is a decreasing sequence of non-empty compact sets with non-empty intersection X := ⋂_{n} Fix(f_{1/n!}). Since any rational number 0 ≠ q ∈ ℚ_{+} can be written in the form q = m/n! for suitable integers m, n > 0, it follows even that ⋂_{q∈ℚ_{+}} Fix(f_{q}) = X ≠ ∅. Consider any parameter t > 0 and any point x ∈ X. Given any sequence q_{1}, q_{2}, … ∈ ℚ_{+} converging to t, the continuity of the orbit t ↦ f_{t}(x) ensures that x = lim_{n} f_{q_{n}}(x) = f_{t}(x).

Lemma 2.2. Let K be a domain in a Banach space E and let f_{t} : D_{t} → E (t ∈ ℝ_{+}] be a family of holomorphic maps defined on open neighborhoods of \overline{K} such that the restrictions [f_{t}|K : t ∈ ℝ_{+}] form a strongly continuous one-parameter semigroup. Assume that for every boundary point x ∈ ∂K there exists a 1-dimensional complex disc Δ_{x} centered in x and intersecting K such that Δ_{x} ⊂ ⋂_{t∈[0,δ_{x}]} D_{t} and ⋃_{t∈[0,δ_{x}]} f_{t}(Δ_{x}) is a bounded set for some δ_{x} > 0. Then [f_{t}|\overline{K} : t ∈ ℝ] is also a strongly continuous one-parameter semigroup.

Proof. By assumption f_{s}(f_{t}(x)) = f_{s+t}(x) (x ∈ K, s, t ∈ ℝ_{+}). Since the maps f_{t}|\overline{K} are all continuous, and it follows f_{s}(f_{t}(x)) = f_{s+t}(x) (x ∈ \overline{K}, s, t ∈ ℝ_{+}) that is [f_{t}|\overline{K} : t ∈ ℝ_{+}] is a one-parameter semigroup of continuous maps on the closure \overline{K}. Hence, to complete the proof, it suffices to see only that for any x ∈ ∂K, the function t ↦ f_{t}(x) is continuous on some neighborhood the origin, namely on [0, δ_{x}). Fix any x ∈ ∂K and consider a convergent sequence t_{n} → t within [0, δ_{x}]. We show the convergence f_{t_{n}}(x) → f_{t}(x) as follows. We can write Δ_{x} = {x+ζv : |ζ| < 1} with a suitable vector v ∈ E. By assumption, the functions g_{n}(ζ) := f_{t_{n}}(x + ζv) (n = 1, 2, …) are uniformly bounded and holomorphic on the unit disc. Furthermore they are assumed to converge pointwise to g(ζ) := f_{t}(x+ζv) on the non-empty open complex domain {ζ ∈ ℂ : x + ζv ∈ K}. In Banach space setting, pointwise convergence implies uniform convergence on compact sets for holomorphic maps [13]. In particular, we have uniform convergence for [g_{n} : n = 1, 2, …] on some compact disc with positive radius. By a theorem of Vigué [18, 9], for a uniformly bounded sequence of holomorphic maps, the uniform convergence on some subdomain entails locally uniform (and hence pointwise) convergence on the whole domain. In particular, g_{n} → g pointwise and hence f_{t_{n}}(x) = g_{n}(0) → g(0) = f_{t}(x).

Corollary 2.3. If E is a JB*-triple, K is its open unit ball and [f_{t} : t ∈ ℝ] is a strongly continuous one-parameter subgroup of Aut(K) then the maps \overline{f}_{t} obtained with graph closure from the respective f_{t}, form a strongly continuous one-parameter group of maps \overline{K} → \overline{K}.

Proof. It is well-known [11] that we can write f_{t} = M_{f_{t}(0)} ∘ U_{t} with some invertible linear operator U_{t} ∈ L(E) and a so-called Möbius transformation with the fractional linear form x ↦ f_{t}(0) + B_{t}[I + L(x, f_{t}(0)]^{−1}x where B_{t}, L(x, f_{t}(0)) ∈ L(E) and ∥L(x, f_{t}(0))∥ ≤ ∥f_{t}(0)∥∥x∥. In particular each f_{t} extends holomorphically to the ball of radius 1/∥f_{t}(0)∥. Thus the conditions required by the lemma are fulfilled since lim_{t→0} f_{(}0) = 0 by assumption.

Lemma 2.4. Let E_{1}, E_{2} be Banach spaces and suppose [W^{t} : t ∈ ℝ_{+}] is a strongly continuous one-parameter subsemigroup of L(E_{1}) with the infinitesimal generator G. Furthermore

let G_{1} ∈ L(E_{2}, E_{1}), G_{2} ∈ L(E_{1}, E_{2}) with G_{1} := [ \begin{matrix}G & 0 \\ G_{1} & 0 \end{matrix} ] and G_{2} := [ \begin{matrix}G & G_{2} \\ 0 & 0 \end{matrix} ]. Then

[[ \begin{matrix}W^{t} & 0 \\ G_{1} ∫_{0}^{t} W^{τ} dτ & I \end{matrix} ] : t ∈ ℝ_{+}] , [[ \begin{matrix}W^{t} & ∫_{0}^{t} W^{τ} dτG_{2} \\ 0 & I \end{matrix} ] : t ∈ ℝ_{+}]

are strongly continuous one-parameter subsemigroups of L(E_{1} ⊕ E_{2}) with the infinitesimal generators G_{1}, G_{2}, respectively.

Proof. Since G_{1} is a bounded perturbation of the generator G := G ⊕ 0 of the semigroup [W^{t} ⊕ 0 : t ∈ ℝ_{+}] with dom(G) = dom(G) ⊕ E_{2}, it suffices to see only that

\frac{d}{dt}|_{t=0+} [ \begin{matrix}W^{t} & 0 \\ G_{1} ∫_{0}^{t} W^{τ} dτ & I \end{matrix} ] [ \begin{matrix}x \\ y \end{matrix} ] = G_{1} [ \begin{matrix}x \\ y \end{matrix} ] (x ∈ dom(G), y ∈ E_{2})

which is immediate. The case of G_{2} is analogous.

Lemma 2.5. Assume [U^{t} : t ∈ ℝ_{+}] is a strongly continuous one-parameter semigroup formed by isometries of a Hilbert space H. Then there is Hilbert space Ĥ containing H as a subspace along with a strongly continuous one-parameter group [Û^{t} : t ∈ ℝ] formed by isometries of Ĥ such that U^{t} = Û^{t}|H (t ≥ 0).

Proof. This is implicit in [15, III. Thm. 9.3].

Lemma 2.6. Assume [U^{t} : t ∈ ℝ_{+}] is a strongly continuous one-parameter semigroup of (necessarily linear) isometries of a Banach space E, E is a subspace of another Banach space F and U^{t_{0}} is the restriction of a surjective isometry of Ê for some t_{0} > 0. Then there is subspace Ê of F along with a strongly continuous one-parameter group [Û^{t} : t ∈ ℝ] of surjective isometries of Ê such that U^{t} = Û^{t}|E (t ≥ 0).

Proof. We may assume t_{0} = 1 without loss of generality and let V be a surjective isometric continuation of U^{1} to F. Notice that V^{−1} a surjective F-isometry with E ⊂ V^{−1}E and hence, in general, V^{n}E ⊃ V^{n+1}E (n ∈ ℤ). Observe also that we have

(2.7) V^{n}U^{t}x = U^{s}V^{m}x whenever V^{m}x ∈ E, n + t = m + s with t, s ≥ 0 and n, m ∈ ℤ.

Hence the definitions

\begin{matrix}Ê_{0} := ⋃_{n=0}^{∞} V^{−n}E = \{x ∈ F : ∃ n ∈ ℤ V^{n}x ∈ E\}, \\ Û_{0}^{t}x := [U^{s}V^{m}x : m ∈ ℤ, s ≥ 0, m + s = t, V^{m}x ∈ E] (x ∈ Ê_{0}) \end{matrix}

make sense. It is also immediate that each operator Û_{0}^{t} is an isometric extension of U^{t} to the not necessarily closed subspace Ê_{0} and, as direct a consequence of (2.7), they have the one-parameter group property Û_{0}^{t}Û_{0}^{t} = Û_{0}^{t+s} (s, t ∈ ℝ). Thus taking the closure of Ê_{0} in F for Ê and setting the graph closure of Û_{0}^{t} for Û^{t} suit the requirements of the lemma.

3. One-parameter semigroups of Caratéodory isometries of the Hilbert ball

Throughout this work H denotes an arbitrarily fixed complex Hilbert space with the scalar product ⟨·|·⟩ and canonical norm ∥x∥ := ⟨x|x⟩^{1/2}. We also keep fixed the standard notations B := {x ∈ H : ∥x∥ < 1} , a^{∗} := [x ↦ ⟨x|a⟩] and I := Id_{H} for its open unit ball, for the adjoint representation of bounded linear functionals and the unit operator. Furthermore, given any subset or single element X of H, we shall write P_{X} for the orthogonal projection of H onto the closed (complex) subspace \overline{Span}(X) spanned by X. Later on we shall regard the elements h and their adjoints h^{∗} of H as column resp. row matrices. In particular, given any vector c ∈ H, we can write P_{c} = [orthogonal projection H → ℂc] = ∥c∥^{−2}cc^{∗} : x ↦ ⟨x|c⟩c x ⊕ ξ ≡ [{x \atop ξ}] (x ∈ H, ξ ∈ ℂ)

L(H ⊕ ℂ) ≡ \{[{A b \atop c^{∗} d}] : A ∈ L(H), b, c ∈ H, d ∈ ℂ\}

In general, if A : S_{1} → H, c : S_{2} → ℂ linear maps, b ∈ H, d ∈ ℂ and S is a linear submanifold in H ⊕ ℂ, A := [{A b \atop c^{∗} d}]|S is the linear map x ⊕ ξ ↦ (Ax + b) ⊕ (c^{∗}x + d) with dom(A) := {x ⊕ ξ ∈ S : x ∈ S_{1} ∩ S_{2}}.

F (A) := [x ↦ (c^{∗}x + d)^{−1}(Ax + b)] with dom(F (A)) = \{x : c^{∗}x + d ≠ 0\} for A = [{A b \atop c^{∗} d}]

T_{x} := [{I x \atop 0 1}] (x ∈ H). Notice: T_{x}^{−1} = T_{−x}.

For c ∈ B, β(c) := \sqrt{1 − ∥x∥^{2}} and Q_{c} := P_{c} + β(c)\overline{P}_{c} with \overline{P}_{c} := I − P_{c}

M_{c} := β(c)^{−1}[{Q_{c} c \atop c^{∗} 1}] lifted Möbius transformation. Notice: M_{−c} = M_{c}^{−1}. Θ_{c} := [x ↦ (1 + ⟨x|c⟩)^{−1}(Q_{c}x + c)] = F (M_{c}) Möbius transformation, Θ_{−c} = Θ_{c}^{−1}.

In 1987 Vesentini in the pioneering works [16,17] established that the mapping

F^{#} : [U^{t} : t∈ℝ_{+}] ↦ [F (U^{t})|B : t∈ℝ_{+}]

constitutes a one-to-one correspondence between the family 𝔘 of all strongly continuous one-parameter semigroups of ℂ-linear isometries of the skew metrics ℓ := ∥x∥^{2} − |ξ|^{2} on H ⊕ ℂ and the family 𝔉 of all strongly continuous one-parameter semigroups [Ψ^{t} : t∈ℝ_{+}] formed by holomorphic Caratéodory isometries of the open unit ball B. Given [U^{t} : t ∈ ℝ_{+}] ∈ 𝔘 with the infinitesimal generator A, for the corresponding non-linear objects Ψ^{t} := F (U^{t}) we have \{p ∈ B : t ↦ Ψ^{t}(p) is differentiable\} = \{x ∈ B : [{x \atop 1}] ∈ dom(A)\} which is dense in the ball B. This observation gives rise to the concept of infinitesimal generator for the the non-linear semigroup [Ψ^{t} : t ∈ ℝ_{+}] = [F^{#}U^{t} : t ∈ ℝ_{+}] as the vector field

Γ(x) := \frac{d}{dt}|_{t=0+}F U^{t}[{x \atop 1}] = −[A[{x \atop 1}]]_{ℂ}x + [A[{x \atop 1}]]_{H}

defined for the points x ∈ dom(Γ) := \{x ∈ B : [{x \atop 1}] ∈ dom(A)\}. The orbit t ↦ Ψ^{t}(0) is differentiable if and only if the linear generator A has the form

A = [ \begin{matrix}iR + νI & b \\ b^{∗} & ν \end{matrix} ]

with some suitable constant ν ∈ ℂ, a vector r ∈ H and a maximal symmetric, in H densely defined not necessarily bounded linear operator R such that dom(A) = dom(R) ⊕ ℂ. In this case the operator iR is the generator of a strongly continuous semigroup of (linear) isometries of H. Choosing any point c ∈ dom(Γ), the orbit t ↦ Φ^{t}(0) with the compositions Φ^{t} := Θ_{−c}∘Ψ^{t}∘Θ_{c} is differentiable. Thus any strongly continuous one-parameter semigroup [Ψ^{t} : t ∈ ℝ_{+}] ∈ 𝔉 is equvialent up to a Möbius transformation (Möbius equivalent for short in the sequel) to a semigroup [Φ^{t} : t ∈ ℝ_{+}] ∈ 𝔉 whose infinitesimal generator has the form

(3.1)  Γ(x) = b − ⟨x|b⟩x + iRx, x ∈ dom(R) ∩ B

with some maximal symmetric operator R defined densely on H and some vector r ∈ H. Also conversely, if R is the infinitesimal generator for some strongly continuous oneparameter subsemigroup of L(H) then, for any b ∈ H, the vector field (3.1) is the infinitesimal generator of a strongly continuous one-parameter subgroup of holomorphic Carathéodory isometries of the ball B.

In [2, 11], the infinitesimal generators of uniformly continuous one-parameter groups of holomorphic Carathéodory isometries on bounded symmetric domains are described completely even in the setting of Banach spaces: up to a biholomorphic transformation, they are equivalent to transformation groups obtained by integration of vector fields of the form x ↦ b − {xb^{∗}x} + iRx with some operation {xy^{∗}z} in 3-variables and a bounded linear hermitian operator over the underlying space. In particular in the case of the unit ball of a Hilbert space we have to apply the triple product \{xy^{∗}z\} = \frac{1}{2}⟨x|y⟩z + \frac{1}{2}⟨z|y⟩x, giving back the spacial case of Vesentini’s formula (3.1). In [11, Sec.2] even a Möbius type explicit formula is given for the integration of the vector fields x ↦ r − {xr^{∗}x}, however the method there seems to be unsuitable in treating the general case with non-vanishing R.

In this paper we develop a triangularization method resulting in an explicit formula for the vector fields (3.1) in terms of fixed points Γ and quadratures of strongly continuous one parameter semigroups of isometries of a 1-codimensional subspace of H. As a consequence we conclude that every strongly continuous one-parameter semigroup of holomorphic Carathódory isometries of B admits a dilation to a strongly continuous one-parameter semigroup of surjective holomorphic Carathódory isometries of the unit ball of some covering Hilbert space. Our results seem to be new even in finite dimensions.

4. Main results

Recall [16] that the continuous extensions of the maps in any Abelian family of holomorphic Carathéodory isometries of B admit common fixed points. Since, in general, the holomorphic Carathéodory isometries of a bounded balanced domain leaving the origin fixed are necessarily linear, any strongly continuous one-parameter semigroup of holomorphic Caratódory isomeries of B is Möbius equivalent to a strongly continuous one-parameter semigroup of linear isometries of H. If there is no common fixed point within the open unit ball then [16] the family of common fixed points reduces to a singleton or two points in the boundary ∂B = {x ∈ H : ∥x∥ = 1}. The following explicit description up to Möbius equivalence of these cases will be our main goal in this paper.

Assumptions, standard notations 4.1. Henceforth, throughout the paper

(i) [Φ^{t} : t ∈ ℝ_{+}] denotes an arbitrarily fixed strongly continuous one-parameter semigroup of holomorphic Carathéodory isometries of B with the infinitesimal generator (3.1); (ii) \overline{x} ∈ ⋂_{t∈ℝ_{+}}Fix(Φ^{t}), ∥\overline{x}∥ = 1; we write P := P_{ℂ\overline{x}}, P_{0} := I −P, H_{0} := P_{0}H = H⊖(ℂ\overline{x});

Theorem 4.2. We have \overline{x} ∈ dom(R) and the operator iP_{0}R|H_{0} is the infinitesimal generator of some strongly continuous one-parameter semigroup [U_{0}^{t} : t ∈ ℝ_{+}] of H_{0}-isometries.

There exist constants λ, μ ∈ ℝ such that, by setting

S := R − iμI, S_{0} := P_{0}S|H_{0}, V_{0}^{t} := e^{iμt}U_{0}^{t}, b_{0} := P_{0}(iSb),

for every point x ∈ B with x_{0} := P_{0}x and P x = ξ\overline{x} we can write

(4.3) \begin{aligned}P Φ^{t}(x) & = [φ_{λ,μ}(t, x_{0}, ξ)^{−1}(ξ − 1)e^{−2λt} + 1]\overline{x} \\ P_{0}Φ^{t}(x) & = φ_{λ,μ}(t, x_{0}, ξ)^{−1}[e^{−λt}V_{0}^{t}x_{0} − (ξ − 1)e^{−2λt}(∫_{0}^{t}e^{λτ} V_{0}^{τ} dτ)b_{0}] where \\ φ_{λ,μ}(t, x_{0}, ξ) & := ⟨(∫_{0}^{t}e^{−λτ} V_{0}^{τ} dτ)x_{0}|b_{0}⟩−(ξ−1)⟨(∫_{0}^{t}e^{−2λτ} ∫_{0}^{τ}e^{λσ}V_{0}^{σ} dσ dτ)b_{0}|b_{0}⟩+ \\ & + (ξ − 1)(λ + iμ)e^{−2λt} + 1. \end{aligned}

Corollary 4.4. Each strongly continuous one-parameter semigroup [Ψ^{t} : t ∈ ℝ_{+}] of holomorphic Carathéodory isometries of B with exactly two joint boundary fixed points is Möbius equivalent to some semigroup [Φ^{t} : t ∈ ℝ_{+}] of the form

(4.5)  \begin{aligned}P Φ^{t}(x) & = \frac{(1 − λ)(1 − ξ)e^{−2λt} + 1}{1 − (1 − ξ)λe^{−2λt}} \overline{x} \\ P_{0}Φ^{t}(x) & = \frac{e^{−λt}}{1 − (1 − ξ)λe^{−2λt}} V_{0}^{t}x_{0} . \end{aligned}

Remark 4.6. (i) The following converse is straightforward to discover from the proof later on. Given any couple of vectors \overline{x}, b_{0} ∈ H such that ∥\overline{x}∥ = 1 and b_{0} ⊥ \overline{x} along with any strongly continuous one-parameter group [V_{0}^{t} : t ∈ ℝ_{+}] of linear isometries of H_{0} = H ⊖ (ℂ\overline{x}) and two real constants λ, μ, the maps (4.3) form a strongly continuous one-parameter group of Carathéodory isometries of B. (ii) It seems that no analogous Möbius equivalent simplification as (4.5) is available in the case of a unique common fixed point lying in the boundary of the unit ball. (iii) In case of λ ≠ 0, it is possible to express the integrated operators in (4.3) in terms of the resolvents R(±λ, iS_{0}). The case λ = 0 is available hence by passing to limits for λ → 0. We can avoid the technical difficulties of this limiting process by means of unitary dilations. Recall that, as direct a consequence of a far reaching result of Deddens [3,Main Thm.]^{∗}, every strongly continuous one parameter semigroup [U^{t} : t ∈ ℝ_{+}] of isometries of H admits a unitary group dilation in the following sense: there exists a Hilbert space Ĥ containing H as a subspace along with a strongly continuous one-parameter group [Û^{t} : t ∈ ℝ] of unitary operators of Ĥ such that U^{t} = Û^{t}|H (t ∈ ℝ_{+}).

Applying a unitary dilation [V̂_{0}^{t} : r ∈ ℝ] of the isometry semigroup [V_{0}^{t} : t ∈ ℝ_{+}] in (4.3), from Remark 4.6(i) we readily obtain the following result with non-linear dilations.

Theorem 4.7. Given any strongly continuous one-parameter semigroup [Ψ^{t} : t ∈ ℝ_{+}] of holomorphic Caratéodory isometries of B, there is a strongly continuous one parameter

∗ For an elementary proof see Section 6. Appendix. group [Ψ̂^{t} : t ∈ ℝ_{+}] of surjective holomorphic Caratéodory isometries of the unit ball B̂ of some Hilbert space Ĥ containing H as a subspace such that Ψ^{t} = Ψ̂^{t}|B (t ∈ ℝ_{+}).

Using the functional calculus of the skew self-adjoint generator iŜ_{0} for the dilation group [V̂_{0}^{t} : t ∈ ℝ] of the semigroup [V_{0}^{t} : t ∈ ℝ_{+}] in the setting of Theorem 4.7, we get following.

Corollary 4.8. In (4.3) we can write

φ_{λ,μ}(t, x_{0}, ξ) = ⟨x_{0}|f_{1}(t, −λ, −Ŝ_{0})b_{0}⟩−(ξ−1)⟨(f_{2}(t, λ, Ŝ_{0})b_{0}|b_{0}⟩+
+ (ξ − 1)(λ + iμ)e^{−2λt} + 1,
P_{0}Φ^{t}(x) = φ_{λ,μ}(t, x_{0}, ξ)^{−1}[e^{−λt} exp(itŜ_{0})x_{0} − (ξ − 1)e^{−2λt}f_{1}(t, λ, Ŝ_{0})b_{0}].

with the bounded analytic functions f_{j}(t, λ, ·) : ℝ → ℂ (j = 1, 2; λ, t ∈ ℝ)

f_{1}(t, λ, s) := ∫_{0}^{t} e^{−λτ} e^{−isτ} dτ = \frac{e^{−t(λ+is)} − 1}{λ + is} = t ∑_{k=0}^{∞} \frac{(−1)^{k+1}}{(k + 1)!} [t(λ + is)]^{k},

f_{2}(t, λ, s) := ∫_{0}^{t} e^{−2λt}f_{1}(τ, λ, s) dτ = \frac{e^{−2λt}(λ − is) − 2λe^{−t(λ−is)} + is}{2λ(λ^{2} + s^{2})(λ − is)} =

= ∑_{n=0}^{∞} [ ∑_{k=0}^{n} \frac{(−2λ)^{n−k}}{(n − k)!} \frac{(λ + is)^{k}}{(k + 1)!} ]\frac{t^{m+2}}{m + 2}.

Example 4.8. Consider the space H := L^{2}(0, ∞) with the semigroup [U^{t} : t ∈ ℝ_{+}] of the isometric shifts U^{t}f := 1_{(0,t)}f([x − t]_{+}). It is well-known that its generator is the (skew symmetric) differential operator iR := d/dx with dom(R) ⊃ C_{0}^{1}(0, ∞). Given any constant ν ∈ ℂ, with the functions

\overline{x} := xe^{−x}, b := (ν − iR)\overline{x} = [(ν + 1)x − 1]e^{−x}

the vector field Γ(f) := b⟨f|b⟩f + iRf (f ∈ dom(R)) gives rise to a strongly continuous one-parameter semigroup Φ^{t} : t ∈ ℝ_{+}] of Carathéodory isometries of the unit ball B such that \frac{d}{dt} |_{t=0+}Φ^{t}(f) = Γ(f) (f ∈ dom(R)) and \overline{x} ∈ ⋂_{t∈ℝ_{+}} Fix(Φ^{t}). We can calculate the isometries U_{0}^{t} in the main Theorem in terms of shifts by the aid of the following observation: The subspace H_{1} := H_{0} ∩[e^{−x}]^{⊥} is [U^{t} : t ∈ ℝ_{+}]-invariant. In particular U_{0}^{t}|H_{1} = U^{t}|H_{1}. Indeed, if f ⊥ xe^{−x}, e^{−x} then for any t > 0 and α, β ∈ ℂ we have

\begin{matrix}⟨U^{t}f|(αx + β)e^{−x}⟩ = ∫_{0}^{∞} 1_{(0,t)}(x)f([x − t]_{+})(αx + β)e^{−x}dx = \\ = ∫_{0}^{∞} f(y)[α(y + t) + β]e^{−(y+t)}dy = αe^{−t}⟨f|xe^{−x}⟩ + (αt + β)e^{−t}⟨f|e^{−x}⟩ = 0 \end{matrix}

Instead of using the orthogonal decomposition H_{0} = H_{1} ⊕ [ℂh_{0}] with h_{0} := (x − 1)e^{−x} ∉ dom(R), it is convenient to pass to matrices associated with the direct sum decomposition H_{0} = H_{1} +^{.} [ℂh_{1}] e.g. with a function h_{1} ∈ dom(R) e.g. with h_{1} := (3x − 2x^{2})e^{−x}. Since iRh_{1} = \frac{d}{dx}h_{1} = (2x^{2} − 7x + 3)e^{−x} and P_{0}f = f − ⟨f|xe^{−x}⟩∥xe^{−x}∥^{−2}xe^{−x} (f ∈ H), we have P_{0}(iRh_{1}) = (2x^{2} − 6x + 3)e^{−x} = g +^{.} 2h_{1} with g := (6x^{2} − 12x + 3)e^{−x} ∈ H_{1}. Thus the H_{1} +^{.} [ℂh_{1}] matrix form of P_{0}(iR)|H_{0} is [{[iR|H_{1}] \atop 0} {g \atop 2}]. Hence, by Lemma 2.4 we get

U_{0}^{t}(f +^{.} ζh_{1}) = [U^{t}f + ζ ∫_{0}^{t} e^{−2τ} U^{τ} g dτ] +^{.} [ζe^{2t}h_{1}] (f ∈ H_{1}, ζ ∈ ℂ).

5. Proofs

Proof. The case (i) is completely treated in [16]: (i) holds if and only if ⋂_{t∈ℝ_{+}} Fix(Ψ^{t}) ≠ ∅

because given any joint fixed point \overline{x} ∈ ⋂_{t∈ℝ_{+}} Fix(Ψ^{t}) [Θ\overline{x} ∘ Ψ^{t} ∘ Θ_{−\overline{x}} : t ∈ ℝ_{+}] is a strongly

continuous one-parameter semigroup of Carathéodory isometries of B leaving the origin fixed which are known to be restrictions of linear isometries to B.

To prove (ii), let ⋂_{t∈ℝ_{+}} Fix(Ψ^{t}) = ∅. As remarked in [16], choosing any point c ∈ B such

that the orbit t ↦ Ψ^{t}(c) is differentiable, the maps Φ^{t} := Θ_{−c} ∘ Ψ^{t} ∘ Θ_{c} form a strongly continuous one-parameter semigroup of Carathódory isometries of B with differentiable 0-orbit. In this case we can write

Φ^{t} = F (U^{t}) (t ∈ ℝ_{+})

with a strongly continuous one-parameter subsemigroup [U^{t} : t ∈ ℝ_{+}] of L(H ⊕ ℂ) whose generator has the matrix form (with respect to the decomposition H ⊕ ℂ)

R = [ \begin{matrix}iR & b \\ b^{∗} & 0 \end{matrix} ] , dom(R) = dom(R) ⊕ ℂ

where R is a densely defined closed symmetric operator on H such that iR is the infinitesimal generator of some strongly continuous one-parameter subsemigroup of isometries in L(H) and b ∈ H. It is also well-known [16] that the continuous extensions \overline{Φ}^{t} of the maps Φ^{t} to the closed unit ball \overline{B} are well-defined and admit one or two common fixed points lying in the boundary ∂B = {x : ∥x∥ = 1}. Henceforth let

\overline{x} ∈ ∂B ∩ ⋂_{t∈ℝ_{+}}Fix(\overline{Φ}^{t}), H_{0} := H ⊖ (ℂ\overline{x}), P := P_{ℂ\overline{x}} = \overline{x} \overline{x}^{∗}, P_{0} := P_{H_{0}} = I − P.

Necessarily \overline{x}⊕1 = [\overline{x} ; 1] is a joint eigenvector of the linear operators U^{t} and for some ν ∈ ℂ,

(5.1)  \overline{x} ∈ dom(R), R[\overline{x} ⊕ 1] = ν[\overline{x} ⊕ 1], b = ν\overline{x} − iR\overline{x}, ν = ⟨\overline{x}|b⟩

implying

(5.1′)  ⟨\overline{x}|R\overline{x}⟩ = 2 Im(ν), b = b_{0} + \overline{ν} \overline{x} where b_{0} := P_{0}b = −iP_{0}R\overline{x}.

Applying the projective shift T : x ⊕ ξ ↦ (x + ξ\overline{x}) ⊕ ξ, consider the operators

V^{t} := e^{−νt}T^{−1}U^{t}T , A := R − ν(I ⊕ 1), B = T^{−1}AT .

Both A and B are bounded perturbations of the operator (iR)⊕1 with dom(B) = dom(A) = dom(R)⊕ℂ. Here A is the infinitesimal generator of the strongly continuous one-parameter subgroup [e^{−νt}U^{t} : t ∈ ℝ_{+}] of L(H ⊕ ℂ). Since T is a bounded invertible linear operator, [V^{t} : t ∈ ℝ] is also a strongly continuous one-parameter semigroup in L(H ⊕ ℂ) with B as infinitesimal generator. Straightforward H ⊕ ℂ-matrix calculation with (5.1) shows that

B = [ \begin{matrix}I & −\overline{x} \\ 0 & 1 \end{matrix} ] [ \begin{matrix}iR − νI & b \\ b^{∗} & −ν \end{matrix} ] [ \begin{matrix}I & \overline{x} \\ 0 & 1 \end{matrix} ] = [ \begin{matrix}B & 0 \\ b^{∗} & 0 \end{matrix} ] where B := iR − νI − \overline{x}b^{∗}.

As being a bounded perturbation of iR, the operator B is the infinitesimal generator of a (unique) strongly continuous semigroup in L(H) which we denote by [V^{t} : t ∈ ℝ_{+}] in the sequel. According to Lemma 2.4(i) we have

(5.2)  V^{t} = [ \begin{matrix}V^{t} & 0 \\ b^{∗}∫_{0}^{τ} V^{τ} dτ & 1 \end{matrix} ]  (t ∈ ℝ_{+}).

Since F (e^{−νt}U^{t}) = F (U^{t}) = Φ^{t} (t ∈ ℝ_{+}), in terms of the translation

T x := x + \overline{x} (x ∈ H),

we can interpret the semigroup [V^{t} : t ∈ ℝ^{+}] as the linear representation by means of F of the semigroup [T^{−1} ∘ Φ^{t} ∘ T : t ∈ ℝ_{+}] formed by holomorphic isometries of the shifted ball B − \overline{x} whose continuous extensions leave the origin fixed. Namely, due to the projective identities F (T^{−1}VT ) = T^{−1}F (V)T (V ∈ L(H ⊕ ℂ)), for the points x ∈ T^{−1}B = B − \overline{x} we have

F (V^{t})x = F (T^{−1}e^{−νt}U^{t}T )x = [T^{−1} ∘ Φ ∘ T]x = Φ(x + \overline{x}) − \overline{x}.

As a consequence of (5.2) and since V^{t}(x ⊕ 1) = [V^{t}x] ⊕ [ ∫_{0}^{t}⟨V^{τ} x| b⟩dτ + 1], we get

(5.3)  Φ^{t}(x) = \frac{V^{t}(x − \overline{x})}{⟨∫_{0}^{t}V^{τ} (x − \overline{x}) dτ|b⟩ + 1} + \overline{x}  (x ∈ B, t ∈ ℝ_{+}).

We proceed to a quadrature formula of V^{t} in terms of [H_{0} ⊕ ℂ\overline{x}]-matrices and isometry semigroups. Consider the [H_{0} ⊕ℂ\overline{x}]-matrix form of its generator B = iR −νI −\overline{x}b^{∗}. Since R is a symmetric operator and b = (νI − iR)\overline{x}, for any vector z ∈ dom(B) = dom(R) we have b^{∗}z = ⟨\overline{ν}z + iRz|\overline{x}⟩ implying also [\overline{x}b^{∗}]z = ⟨z|ν\overline{x} − iR\overline{x}⟩\overline{x} = P (\overline{ν}I + iR)z. Thus, by setting

λ := Re(ν), μ := Im(ν) = ⟨\overline{x}|R\overline{x}⟩/2, S := R − μI

we can write b^{∗} = \overline{x}^{∗}(λI + iS), \overline{x}b^{∗} = P (λI + iS), B = iR − νI − \overline{x}b^{∗} = P_{0}[iS] − λ(I + P). Furthermore, by (5.1) and (5.1′), B\overline{x} = (iS − λI)\overline{x} − ⟨\overline{x}|b⟩\overline{x} = −b − ν\overline{x} = −[b_{0} ⊕ (2λ\overline{x})]. Thus, in terms of H_{0} ⊕ (ℂ\overline{x})]-matrices,

B =[ \begin{matrix}P_{0}[iS−λI]|H_{0} & iP_{0}S\overline{x} \\ 0 & −2λ \end{matrix} ]=[ \begin{matrix}iS_{0}+λI_{0} & −b_{0} \\ 0 & 0 \end{matrix} ]−2λ[ \begin{matrix}I_{0} & 0 \\ 0 & 1 \end{matrix} ] where \begin{matrix}S_{0} := P_{0}SP_{0}|H_{0}, \\ I_{0} := Id_{H_{0}}. \end{matrix}

Since \overline{x}∈dom(S) and P_{0}SP_{0} =S−P S−SP +P SP =R−[μI+\overline{x}(S\overline{x})^{∗}+(S\overline{x})\overline{x}^{∗}−⟨\overline{x}|S\overline{x}⟩\overline{x}\overline{x}^{∗}], the operator P_{0}[iS]P_{0} is a bounded perturbation of the infinitesimal generator iR. Hence its restriction iS_{0} to H_{0} is also the infinitesimal generator a strongly continuous semigroup [V_{0}^{t} : t ∈ ℝ_{+}] of isometries on range(\overline{P}) = H_{0}. By Lemma 2.4(ii) it follows

V^{t} = e^{−2λt} [ \begin{matrix}e^{λt}V_{0}^{t} & ∫_{0}^{t}e^{λτ} V_{0}^{τ} b_{0} dτ \\ 0 & 1 \end{matrix} ] (t ∈ ℝ_{+})

in terms of H_{0} ⊕ (ℂ\overline{x})-matrices. Therefore, for any x = x_{0} + (ξ\overline{x}) with x_{0} ∈ H_{0} and ξ ∈ ℂ,

(5.4)  V^{t}(x − \overline{x}) = e^{−λt}V_{0}^{t}x_{0} + e^{−2λt}(1 − ξ)[∫_{0}^{t}e^{λτ} V_{0}^{τ} b_{0} dτ − \overline{x}].

Next we calculate the terms b^{∗}V^{τ} (x−\overline{x}) = ⟨V^{t}(x−\overline{x})| b_{0} +\overline{ν} \overline{x}⟩. Since range(V_{0}^{τ})⊂H_{0} ⊥\overline{x}, we have ⟨V^{τ} (x − \overline{x})| \overline{x}⟩ = −e^{−2λτ} (1 − ξ). Also b_{0} ⊥ \overline{x}. Therefore

\begin{aligned}b^{∗}V^{τ} (x − \overline{x}) & = ⟨V^{τ} (x − \overline{x})|b_{0} + \overline{ν} \overline{x}⟩ = \\ & = e^{−λτ} ⟨V_{0}^{τ} x_{0}|b_{0}⟩ + e^{−2λτ} (1 − ξ)[∫_{0}^{τ}e^{λσ}⟨V_{0}^{σ}b_{0}|b_{0}⟩dσ − ν]. \end{aligned}

Substituting this into (5.3), in terms of integrals of strongly continuous isometry valued functions, we get

P Φ^{t}(x) = [φ(λ, ξ, x_{0})^{−1}(ξ − 1)e^{−2λt} + 1]\overline{x}
P_{0}Φ^{t}(x) = φ(λ, ξ, x_{0})^{−1}[e^{−λt}V_{0}^{t}x_{0} − (ξ − 1)e^{−2λt}(∫_{0}^{t}e^{λτ} V_{0}^{τ} dτ)b_{0}] where
φ(λ, ξ, x_{0}) := ⟨(∫_{0}^{t}e^{−λτ} V_{0}^{τ} dτ)x_{0}| b_{0}⟩ − (ξ − 1)⟨(∫_{0}^{t}e^{−2λτ} ∫_{0}^{τ}e^{λσ}V_{0}^{σ} dσ dτ)b_{0}| b_{0}⟩+
+ (ξ − 1)(λ + iμ)e^{−2λt} + 1.

Remark 5.5. We can eliminate the integrals above in terms of resolvents as follows. As being iS_{0} the generator of the semigroup [V_{0}^{t} : t ∈ ℝ_{0}], the operator S_{0} is closed with dense domain in H_{0}. Since S_{0} is also symmetric, both ±iS_{0} are dissipative (namely Re⟨±iSx_{0}|x_{0}⟩ = 0 for x_{0} ∈ dom(S)) with the properties that both range(±iS_{0} + δI_{0}) are dense in H for any δ > 0 and that the operators (iS +δI_{0})^{−1} : range(S) → H_{0} (0 ≠ δ ∈ ℝ) are all bounded and densely defined.^{∗}

∗ Indeed y_{0} ⊥ range(±iS_{0} +δI_{0}) means 0 = ⟨±iS_{0}x_{0} −δx_{0}|y_{0}⟩ that is 0 = ⟨x_{0}|∓iS_{0}y_{0} −δy_{0}⟩ for (x_{0} ∈ dom(S)) entailing ∓iS_{0}y_{0} + δy_{0} = 0 with δ∥y_{0}∥^{2} = ±i⟨S_{0}y_{0}|y_{0}⟩ ∈ iℝ which is possible only if y = 0. Thus by the Lumer-Phillips theorem [5, II. Thm. 3.15], also the operator −iS_{0} generates a strongly continuous contraction (actually isometry) semigroup and all the values 0 ≠ δ ∈ ℝ belong to the resolvent set of iS.

Given δ ∈ ℝ \ {0}, by [5, II. Lemma 1.3] for any x_{0} ∈ range(iS_{0} − δI_{0}) and t > 0 we have ∫_{0}^{t} e^{−δτ} V_{0}^{τ} x_{0} dτ = ∫_{0}^{t} e^{−δτ} V_{0}^{τ} (iS_{0}−δI_{0})[(iS_{0}−δI_{0})^{−1}x_{0}] dτ = (e^{−δt}V_{0}^{t}−I_{0})(iS_{0}−δI_{0})^{−1}x_{0}. The boundedness of the operators V^{t} and the resolvents R(δ, iS) = closure((δI_{0} − iS_{0})^{−1}) establishes that for all t > 0 and 0 ≠ λ ∈ ℝ,

∫_{0}^{t} e^{−λτ} V_{0}^{τ} dτ = (I_{0} − e^{−λt}V_{0}^{t})R(λ, iS_{0}),

∫_{0}^{t} e^{−2λτ} ∫_{0}^{τ} e^{λσ}V_{0}^{σ} dσ dτ = ∫_{0}^{t} e^{−2λτ} (I_{0} − e^{λτ} V_{0}^{τ} )R(−λ, iS_{0}) dτ =

= \frac{1 − e^{−2λt}}{2λ} R(−λ, iS_{0}) − (I_{0} − e^{−λt}V_{0}^{t})R(λ, iS_{0})R(−λ, iS_{0})

with integrals of strongly continuous bounded operator valued functions. Taking (5.3) and (5.4) into account, hence we conclude that for any x = x_{0} + ξ\overline{x} ∈ B,

\begin{aligned}P Φ^{t}(x) & = [φ(λ, ξ, x_{0})^{−1}(1 − ξ)e^{−2λt} + 1]\overline{x} \\ P_{0}Φ^{t}(x) & = φ(λ, ξ, x_{0})^{−1}[e^{−λt}V_{0}^{t}x_{0} − (1 − ξ)e^{−2λt}(I_{0} − e^{−λt}V_{0}^{t})R(−λ, iS_{0})b_{0}] \end{aligned}

where

φ(λ, ξ, x_{0}) := ⟨(I_{0} − e^{−λt}V_{0}^{t})R(λ, iS_{0})x_{0}| b_{0}⟩ + (1 − ξ)⟨\frac{1 − e^{−2λt}}{2λ} R(−λ, iS_{0})b_{0}| b_{0}⟩−
− (1 − ξ)⟨(I_{0} − e^{−λt}V_{0}^{t})R(λ, iS_{0})R(−λ, iS_{0})b_{0}| b_{0}⟩−
− (1 − ξ)(λ + iμ)e^{−2λt} + 1.

The case of two boundary fixed points

Assume ⋂_{t∈ℝ_{+}} Fix(\overline{Φ}^{t}) = \{\overline{x}_{1}, \overline{x}_{2}\} ⊂ ∂bfB with \overline{x}_{1} ≠ \overline{x}_{2}. Observe that in this case there exists t_{0} with Fix(Φ^{t_{0}}) = \{\overline{x}_{1}, \overline{x}_{2}\}. Indeed, in the contrary case, the whole real segment [\overline{x}_{1}, \overline{x}_{2}] would be contained in Fix(Φ^{t}) for every index t. It is well-known that there exists a Möbius transformation Θ = F (M) and a vector \overline{x} ∈ ∂B such that Fix(Θ ∘ \overline{Φ}^{t_{0}} ∘ Θ^{−1}) = \{Θ(\overline{x}_{1}), Θ(\overline{x}_{1})\} = \{±\overline{x}\}. It follows ⋂_{t∈ℝ_{+}} Fix(Θ ∘ \overline{Φ}^{t} ∘ Θ^{−1}) = \{±\overline{x}\}. Since the or-

bits t ↦ \overline{Φ}^{t}(±\overline{x}) are differentiable (actually constants), also the orbit of 0 = −\overline{x} + \overline{x} is differentiable. In particular, (±\overline{x}) ⊕ 1 are joint eigenvectors of the linear representation operators MU^{t}M^{−1} with F (U^{t}) = Φ^{t}. Thus, by passing to the Möbius equivalent semigroup [Θ ∘ \overline{Φ}^{t} ∘ Θ^{−1} : t ∈ ℝ_{+}] instead of [\overline{Φ}^{t} : t ∈ ℝ_{+}], in this case we may assume without loss of generality that

⋂_{t∈ℝ_{+}} Fix(\overline{Φ}^{t}) = \{±\overline{x}\}, R[(−\overline{x}) ⊕ 1] = ν_{1}[(−\overline{x}) ⊕ 1], R[\overline{x} ⊕ 1] = ν_{2}[\overline{x} ⊕ 1]

for some constants ν_{1}, ν_{2} ∈ ℂ. Hence from (5.1) we see that b = −(ν_{1}I −iR)\overline{x} = (ν_{2}I −iR)\overline{x} entailing that \overline{x} is an eigenvector of R with eigenvalue \frac{1}{2i}(ν_{1} + ν_{2}) = ⟨R\overline{x}|\overline{x}⟩ ∈ ℝ. Taking also (5.1′) in account we deduce b = 2^{−1}(ν_{2} − ν_{1})\overline{x}, b_{0} = P_{0}b = 0, b = −\overline{ν_{1}}\overline{x} = \overline{ν_{2}}\overline{x}, 2Im(ν_{k}) = ⟨R\overline{x}|\overline{x}⟩ = (2i)^{−1}(ν_{1} + ν_{2}) = 0. Thus, for some λ ∈ ℝ, necessarily ν_{1} = λ, ν_{2} = −λ entailing also R\overline{x} = 0, b = λ\overline{x}, R[(±\overline{x}) ⊕ 1] = ±λ[(±\overline{x}) ⊕ 1]. With the choice ν := ν_{1}, in terms of (5.2) this means that b_{0} = 0, S_{0} = P_{0}RP_{0} = R and hence

\begin{aligned}P Φ^{t}(x) & = \frac{(1 − λ)(1 − ξ)e^{−2λt} + 1}{1 − (1 − ξ)λe^{−2λt}} \overline{x} \\ P_{0}Φ^{t}(x) & = \frac{e^{−λt}}{1 − (1 − ξ)λe^{−2λt}} V_{0}^{t}x_{0} . \end{aligned}

Lemma 5.6. Assume [V^{t} : t ∈ ℝ_{+}] is a strongly continuous one-parameter semigroup in L(H ⊕ ℂ) whose infinitesimal generator is a bounded perturbation of a skew symmetric operator and V^{t}(0⊕1) = 0⊕1 (t ∈ ℝ_{+}). Then there is a strongly continuous one-parameter semigroup [V^{t} : t ∈ ℝ_{+}] in L(H) along with a vector b ∈ H such that

V^{t} = [ \begin{matrix}V^{t} & 0 \\ [W^{t}b]^{∗} & 1 \end{matrix} ] where W^{t}b = ∫_{0}^{t} V^{τ} dτ (t ∈ ℝ_{+}).

In particular, the infinitesimal generator B of [V^{t} : t ∈ ℝ_{+}] has the matrix form

B = \left[{B 0 \atop b^{∗} 0}\right] , dom(B) = dom(B) ⊕ ℂ

where B is the infinitesimal generator of [V^{t} : t ∈ ℝ_{+}].

Proof. Since V^{t}(0 ⊕ 1) = 0 ⊕ 1 (t ∈ ℝ_{+}), the operators V^{t} have the matrix form

V^{t} = [ \begin{matrix}V^{t} & 0 \\ v_{t}^{∗} & 1 \end{matrix} ] , V^{t} ∈ L(H), v_{t} ∈ H (t ∈ ℝ_{+})

with respect to the decomposition H ⊕ ℂ. Considering orbits of the form t ↦ V^{t}[x ⊕ 0] = [V^{t}x] ⊕ ⟨x|v_{t}⟩ we see that [V^{t} : t ∈ ℝ_{+}] is a continuous one-parameter semigroup in L(H). Also 0 ⊕ 1 ∈ dom(B) with B[0 ⊕ 1] = 0. In particular x ⊕ 0 = [x ⊕ ξ] − ξ[0 ⊕ 1] ∈ dom(B) for any vector x ⊕ ξ ∈ dom(B), showing that dom(B) = [X ⊕ 0] + ℂ[0 ⊕ 1] for some dense linear subspace of H. We have x ∈ X if and only if the orbit t ↦ V^{t}[x ⊕ 0] = [V^{t}x] ⊕ ⟨x|v_{t}⟩ is differentiable. Hence we deduce that x ∈ X if and only if the functions t ↦ V^{t}x and t ↦ ⟨x|v_{t}⟩ are simultaneously differentiable. Thus x ∈ X if and only if x ∈ dom(B) and t ↦ ⟨x|v_{t}⟩ is differentiable. .

Henceforth [Ψ^{t} : t ∈ ℝ_{+}] is an arbitrarily fixed strongly continuous one-parameter semigroup of Caratéodory isometries of B.

Remark 5.7. We know that ⋂_{t∈ℝ_{+}} Fix(\overline{Ψ}^{t}) ≠ ∅. By fixing such a common fixed point

\overline{x} ∈ \overline{B} with \overline{Ψ}^{t}(\overline{x}) = \overline{x}, we know also that [[x ↦ Ψ^{t}(x + \overline{x}) − \overline{x}] : t ∈ ℝ_{+}] is a strongly continuous semigroup of maps \overline{K} → \overline{K} with common fixed point 0 for the shifted ball K := −\overline{x} + B. Hence we can write

Ψ^{t} = F (V^{t}) with V^{t} := T \overline{x}U^{t}T_{−\overline{x}} (t ∈ ℝ_{+})

where [U^{t} : t ∈ ℝ_{+}] is a strongly continuous one-parameter semigroup of triangular operators of the form described in Lemma 2.4. As suggested in [16], let us consider a modification

Φ^{t} := Θ_{−c} ∘ Ψ^{t} ∘ Θ_{c} = F (W^{t}), W^{t} := M_{−c}T\overline{x}U^{t}T_{−\overline{x}}M_{c} (t ∈ ℝ_{+})

with a Möbius transformation Θ_{c} such that the orbit t ↦ Φ^{t}(0) = Θ_{−c}(Ψ^{t}(c)) should be differentiable. In terms of the generators A := \frac{d}{dt} |_{t=0}U^{t} and R := \frac{d}{dt} |_{t=0}W^{t} this means the requirement 0 ⊕ 1 ∈ dom(R). In any case,

(5.8)  A = T_{−\overline{x}}M_{c}RM_{−c}T\overline{x} , dom(A) = T_{−\overline{x}}M_{c} dom(R).

In particular the suitable points c with T_{−\overline{x}}M_{c}(0 ⊕ 1) = (c − \overline{x}) ⊕ 1 ∈ dom(R) = dom(A) are dense in B. Recall that with some closed symmetric, in H densely defined operator R, with a vector r ∈ H and some constant ν ∈ ℂ we have

R = ν[I ⊕ 1] + \left[{iR r \atop r^{∗} 1}\right] , dom(R) = X ⊕ ℂ where X := dom(R).

Also, with some densely defined operator A and a densely defined linear functional a,

A = \left[{A 0 \atop a 0}\right] |Z ⊕ ℂ, Z := \{z ∈ H : z ⊕ 0 ∈ domA\} ⊂ dom(A) ∩ dom(a).

Then from (5.8) we conclude that Z ⊕ ℂ = dom(A) = M_{−c}T\overline{x} (X ⊕ ℂ) that is

Z = \{z : z ⊕ 0 ∈ dom(A)\} = \{z : ∃ x ∈ X, ξ ∈ ℂ z ⊕ 0 = M_{−c}T\overline{x} (x ⊕ ξ)\}.

Taking into account the straightforward relations

β(c)T_{−\overline{x}}M_{c} = \left[{Q_{c} − \overline{x}c c − \overline{x} \atop c^{∗} 1}\right] , β(c)M_{−c}T\overline{x} = \left[{Q_{c} Q_{c}\overline{x} − c \atop −c^{∗} 1 − c^{∗}\overline{x}}\right] ,

Q_{c}c = [P_{c} + β(c)(I − P_{c})]c = c, cc^{∗} = [x ↦ ⟨x|c⟩c] = ∥c∥^{2}P_{c},

Q_{c}(Q_{c}−cc^{∗}) = [P_{c}+β(c)\overline{P}_{c}][P_{c}+β(c)\overline{P}_{c}−∥c∥^{2}P_{c}] = β(c)^{2}P_{c}+β(c)^{2}\overline{P}_{c}) = (1 − ∥c∥^{2})I we deduce that

Z = \{(Q_{c} − \overline{x}c^{∗})x + ξ(c − \overline{x}) : x ∈ X, ξ ∈ ℂ, c^{∗}x + ξ = 0\} =
= \{(Q_{c} − \overline{x}c^{∗})x − (c^{∗}x)(c − \overline{x}) : x ∈ X\} = [Q_{c}\overline{x}c^{∗} − (c − \overline{x})c^{∗}]X =
= [Q_{c} − cc^{∗}]X = Q_{c}^{−1}X.

Assumption 5.9. [Ψ^{t}|B : t ∈ ℝ_{+}] is a strongly continuous one-parameter semigroup.

Remark 5.10. We know from [6] the following facts:

(i) Ψ^{t} =F (V^{t}) (t∈ℝ^{+}) with suitable operators V^{t} ∈ L(H ⊕ ℂ). Actually, by writing c_{t} := Ψ^{t}(0), Q_{t} := P_{t} + \sqrt{1 − ∥c_{t}∥^{2}}\overline{P}_{t} with the orthogonal projection P_{t} : H → ℂc_{t}, we can take V^{t} := [{Q_{t} c_{t} \atop c_{t}^{∗} 1} ][{U_{t} 0 \atop 0 1}] where U_{t} is some linear isometry H → H. Later on we need only the following immediate consequences of this representation: (ii) ⋂_{t∈[0,τ]} dom(Ψ^{t}) is a neighborhood of \overline{B} for all τ >0; (iii) Ψ^{t}(∂B) ⊂ ∂B (t ∈ ℝ_{+}).

Lemma. We have ⋂_{t∈ℝ_{+}}Fix(Ψ^{t}) ≠ ∅.

Assumptions. 1) ∥\overline{x}∥ = 1; 2) U^{t} = [{U^{t} 0 \atop u^{t} 1}] (t ∈ ℝ_{+}); 3) dU^{t}/dt|_{t=0} = [{A 0 \atop a 0}] where

4) A = \overline{P} [iS] + λ(I + P), ax = ⟨(iS − λI)x|\overline{x}⟩, 5^{∗}) \overline{x} ∈ dom(A) = dom(S)

with S closed symmetric, P := \overline{x}\overline{x}^{∗} projection onto ℂ\overline{x}, \overline{P} := I − P.

Then

U^{t} = exp (tA), u^{t}x = ∫_{0}^{t} aU^{s}x ds = ∫_{0}^{t} ⟨(iS − λI)x|\overline{U}^{s}x⟩ ds.

With the decomposition H = [\overline{P} H] ⊕ [ℂ\overline{x}],

A = \overline{P} [iS + λI] + 2λP = [ \begin{matrix}\overline{P} [iS + λI]\overline{P} & \overline{P} [iS + λI]\overline{x} \\ 0 & 2λ \end{matrix} ]

The unbounded operator

Y := \overline{P} [iS + λI]\overline{P} , dom(C) = dom(S)

is well-defined, since (according to assumption 5^{∗}) \overline{P} x = x − ⟨x|\overline{x}⟩\overline{x} ∈ domS for every x ∈ dom(S). By writing also

\overline{y} := \overline{P} [iS + λI]\overline{x}

for short, we get

exp(tA) = [ \begin{matrix}exp(tY) & w^{t} \\ 0 & e^{2λt} \end{matrix} ] , dw^{t}/dt = Y w^{t} + e^{2λt}\overline{y} = 2λw^{t} + exp(tY)\overline{y}

as a consequence of the equations \frac{d}{dt}[{exp(tY) w^{t} \atop 0 e^{2λt}} ] = [{Y \overline{y} \atop 0 2λ}][{exp(tY) w^{t} \atop 0 e^{2λt}} ] = [{exp(tY) w^{t} \atop 0 e^{2λt}} ][{Y \overline{y} \atop 0 2λ}]. Due to the skew symmetry of iS, each operator exp(tY) has the form e^{λt} exp(it\overline{P} S\overline{P}) = e^{λt}[isometry] on the subspace \overline{P} H. The initial value problem

\frac{dw^{t}}{dt} = 2λw^{t} + exp(tY)\overline{y}, w^{0} = 0

can be solved explicitly by means of the substitution w^{t} = e^{2λt}u^{t}:

2λe^{2λt}u^{t} + e^{2λt} \frac{du^{t}}{dt} = 2λe^{2λt}u^{t} + e^{λt} exp(itS)\overline{y}, ⇒ u^{t} = ∫_{0}^{t} e^{−λs} exp(is\overline{P} S\overline{P})\overline{y} ds

with the result w^{t} = ∫_{0}^{t} e^{λ(2t−s)} exp(is\overline{P} S\overline{P})\overline{y} ds, whence

exp(tA) = e^{λt} [ \begin{matrix}exp(it\overline{P} S\overline{P}) & ∫_{0}^{t} e^{t−s} exp(is\overline{P} S\overline{P}) ds \\ 0 & e^{λt} \end{matrix} ] .

A = i ∫_{−∞}^{∞} ξ dE(ξ)

U^{t} = exp (t(iA + λ)) = ∫_{−∞}^{∞} e^{t(iξ+λ)} dE(ξ)

u^{t}z = ∫_{0}^{t} aU^{s}z ds; av = ⟨(iA − λ)v|\overline{x}⟩

u^{t}z = ∫_{0}^{t} ⟨(iA − λ)U^{s}z|\overline{x}⟩ =

= ∫_{ξ=−∞}^{∞} ∫_{s=0}^{t}(iξ − λ)e^{t(iξ+λ)} ds d⟨E(ξ)z|\overline{x}⟩ =

= ∫_{ξ=−∞}^{∞} \frac{(e^{t(iξ+λ)})(iξ − λ)}{iξ + λ}d⟨E(ξ)z|\overline{x}⟩

Ψ^{(}x) = \overline{x} + [U^{t}(x − \overline{x}) ⊕ 1]_{ℂ}^{−1}[U^{t}(x − \overline{x}) ⊕ 1]_{H}

U^{t}(z ⊕ ζ) = [U^{t}z] ⊕ [u^{t}z + ζ]

6. Fractional-linear automorphisms of a circular domain in Hilbert space

Notations 6.1. H complex Hilbert space with scalar product ⟨x|y⟩ being linear in x; u_{∗} := [x ↦ ⟨x|u⟩] for any vector u ∈ H; H ⊕ ℂ = \{x ⊕ ξ : x ∈ H, ξ ∈ ℂ\} with x ⊕ ξ := [{x \atop ξ}], (x ⊕ ξ)_{H} := x, (x ⊕ ξ)_{ℂ} := ξ. K bounded convex circular domain in H with center \overline{x} (that is K − \overline{x} = e^{it}[K − \overline{x}] t ∈ ℝ). Ψ^{t}(x) (t ∈ ℝ) injective fractional linear transformations with 0 ∈ Fix(Ψ^{t}). Without loss of generality we shall write

Ψ^{t}(x) := (1 + u_{∗}^{t}x)^{−1}U^{t}x = [U^{t}(x ⊕ 1)]_{ℂ}^{−1}[U^{t}(x ⊕ 1)]_{H}

where U^{t} ∈ L(H), u^{t} ∈ H, U^{t} := [ \begin{matrix}U^{t} & 0 \\ u_{∗}^{t} & 1 \end{matrix} ] .

We identify the matrix U^{t} with the linear operator x ⊕ ξ ↦ U^{t}(x ⊕ ξ).

Assumption 6.2. \overline{K} ⊂ dom(Ψ^{t}) = \{x ∈ H : 1 + u_{∗}^{t}x ≠ 0\} and [Ψ^{t} : t ∈ ℝ] is a strongly continuous one-parameter group in Aut(K) with 0 ∈ \overline{K}.

Remark 6.3. It is immediate that Assumption 6.2 implies that the family [U^{t} : t ∈ ℝ] is a strongly continuous one-parameter subgroup of L(H ⊕ ℂ), in particular

(6.4) U^{t+s} = U^{t}U^{s}, U^{t+s} = U^{t}U^{s}, u_{∗}^{t+s} = u_{∗}^{t}U^{s} + u_{∗}^{s} = [(U^{s})^{∗}u^{t}]_{∗} + u_{∗}^{s}  (t, s ∈ ℝ).

Proof. (6.4) is a straightforward translation of the relation Ψ^{t+s} = Ψ^{t}∘Ψ^{s}. The continuity of the functions t ↦ U^{t}(x ⊕ ξ) = (U^{t}x) ⊕ (u_{∗}^{t}x + ξ) can be deduced from the assumed continuity of the functions t ↦ Ψ^{t}(x) (x ∈ K) as follows. Given 0 ≠ x ∈ K and t_{0} ∈ ℝ arbitrarily, we can choose λ > 1 with λx ∈ K since K is an open set. Since the maps Ψ^{t} are injective (actually invertible), we can also choose a vector v ∈ H with v_{∗}Ψ^{t}(x) ≠ v_{∗}Ψ^{t}(λx) for t in some interval J around t_{0}. Then v_{∗}Ψ^{t}(μx) = [μ/(1 + u_{∗}^{t}x)]v_{∗}U^{t}x that is [μv_{∗}Ψ^{t}(μx)]u_{t}^{∗}x − v_{∗}U^{t}x = −v_{∗}Ψ^{t}(x) for t ∈ J and μ = 1, λ. It follows that u_{∗}^{t}x = [λΨ^{t}(x) − Ψ^{t}(λx)]/[λ(Ψ^{t}(x) − Ψ^{t}(λx))] is a continuous function of t on the interval J. Hence we get the also continuity of t ↦ U^{t}x = (1 + u_{∗}^{t}x)Ψ^{t}(x) establishing the continuity of t ↦ U^{t}(x ⊕ ξ) for any ξ ∈ ℂ.

Acknowledgements. The author’s research was supported by the project TÁMOP-4.1.1.C-12/1/KONV-2012-0005 Impulse Lasers for Use in Materials Science and Biophotonics of the European Union and co-financed by the European Social Fund.

References

[1] M. Abate, Common fixed points of commuting holomorphic maps, Math. Ann. 283 (1989) 645-655.

[3] J.A. Deddens, On extending semigroups of contractions, Trans. Amer. Math. Soc., 145 (1969), 233-239.

[2] H. Braun – W. Kaup – H. Upmeier, On the automorphisms of circular and Reinhardt domains in complex Banach spaces, Manuscripta Math., 25 (1978), 97-133.

[4] S. Dineen, Complete holomorphic vector fields on the second dual of a Banach space, Math. Scand. 59 (1986) 131-142.

[5] K.-J. Engel – R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics Vol. 194, Springer-Verlag, New York, 2000.

[6] T. Franzoni – E. Vesentini, Holomorphic Maps and Invariant Distances, North Holland Math. Studies 40, North Holland Publ. Co., Amsterdam-New York-Oxford, 1980.

[7] K. Goebel, Fixed points and invariant domains of holomorphic mappings of the Hilbert ball, Nonlinear Analysis, Theory, Methods and Applications, 6/12 (1982) 1327-1334.

[8] T.L. Hayden – T.J. Suffridge, Biholomorphic maps in Hilbert space have a fixed point, Pacific J. Math., 38 (1971) 419-422.

[9] J.-M. Isidro – L.L. Stachó, Holomorphic Automorphism Groups in Banach Spaces, North Holland Math. Studies 105, North Holland Publ. Co., Oxford–New York–Amsterdam, 1985.

[10] W. Kaup, Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension I., Math. Ann., 357 (1981), 463–481.

[10] W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z., 83 (1983), 503–529.

[12] V. Katshkevich - S. Reich - D. Shoiket, One-parameter semigroups of fractional-linear transformations, Operator Theory Advances and Applications, 123 (2001) 401–411.

[13] L. Nachbin, Holomorphic functions, domains of holomorphy and local properties, North-Holland Pub. Co., Oxford–New York–Amsterdam, 1970.

[14] F. Riesz – B. Sz.-Nagy, Vorlesungen über Funktionalanalysis, VEB Deutscher Verlag der Wissenschaften, Berlin, 1968.

[15] B. Sz.-Nagy – C. Foias, Harmonic Analysis of Operators on Hilbert Space, Akadmiai Kiad/North-Holland Publ. Co., Budapest–Amsterdam–London

[16] E. Vesentini, Semigrioups of holomorphic isometries, Advances in Mathemetics, 65 (1987), 272–306.

[17] E. Vesentini, Semigrioups of holomorphic isometries, in: P.M. Gauthier (ed.), Complex Potential Theory, pp. 475–548. Kluwer Academic Pulbl., Netherlands, 1994.

[18] J.-P. Vigué, Le group des automorphismes analytiques d′un domaine borné d′un espace de Banach complexe. Applications aux domaines bornés symmetriques, Ann. Sci. Ecole Norm. sup., 9 (1976), 203–282.

L.L. STACHÓ Bolyai Institute University of Szeged Aradi Vértanúk tere 1 H-6720 SZEGED, HUNGARY

E-mail: stacho@math.u-szged.hu